Here is my solution to the problem I posted yesterday.
I would make use of the fact that $D(X)$ is a locally convex algebra. Assuming that one can define a bornology on it (see http://arxiv.org/pdf/0706.0027.pdf) the smash product $D(X)\#G$ can be made into a bornological algebra, too. Then we may use the fact that the Hochschild cohomology of any bornological algebra is defined in terms of the Ext-functor. The rest is as in Etingof's paper http://arxiv.org/pdf/math/0406499v3.pdf.